A SURVEY of OFF-AXIS PARABOLA ALIGNMENT TECHNIQUES by Julia Zugby a Report Submitted to the Faculty of the DEPARTMENT of OPTICA

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A SURVEY OF OFF-AXIS PARABOLA ALIGNMENT TECHNIQUES

Julia Zugby

A Report Submitted to the Faculty of the

DEPARTMENT OF OPTICAL SCIENCES

In Partial Fulfillment of the Requirements

For the Degree of

MASTERS OF SCIENCE

In the Graduate College

THE UNIVERSITY OF ARIZONA

2018

THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

The master’s report title “A Survey of Off-Axis Parabola Alignment Techniques”, prepared by Julia Zugby, has been submitted in partial fulfillment of requirements for a master’s degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

As members of the Master’s Committee, we certify that we have read the report prepared by Julia Zugby, “A Survey of Off-Axis Parabola Alignment Techniques”, and recommend that it be accepted as fulfilling the report requirement for the Master’s Degree.

Dr. Dae Wook Kim, Chair Date

Dr. Rongguang Liang, Member Date

Dr. Jim Schwiegerling, Member Date

APPROVAL BY REPORT ADVISOR

This report has been approved on the date shown below:

Dr. Dae Wook Kim Date

Professor of Optical Sciences

ACKNOWLEDGEMENTS

Thank you.

3 Table of Contents

ABSTRACT ...... 7

1 INTRODUCTION ...... 8

1.1 MOTIVATION ...... 8

1.2 THE OAP ...... 8

1.3 ABERRATIONS OF THE OAP...... 10

1.4 SPECIFYING THE OAP ...... 14

2 GENERAL ALIGNMENT ...... 15

2.1 XYZ COORDINATE MEASURING INSTRUMENTS ...... 17

2.1.1 The Laser Tracker ...... 18

2.1.2 The Coordinate Measurement Machine (CMM) ...... 21

2.2 THE INTERFEROMETER ...... 23

3 OAP SPECIFIC ALIGNMENT TECHNIQUES ...... 25

3.1 TWO LASER TECHNIQUE ...... 25

3.2 LASER TRACKER TECHNIQUE ...... 29

3.3 COMPUTER AIDED TECHNIQUE USING ZERNIKES ...... 33

3.4 SUMMARY...... 35

4 CASE STUDY ...... 36

5 CONCLUSION ...... 38

APPENDIX A – NOLL ZERNIKE EXPANSION ...... 39

REFERENCES ...... 40

4 LIST OF FIGURES

Figure 1: OAP diagram from Optikos [1]...... 9

Figure 2: System diagram, pupil map, and ray-fan for an arbitrary OAP as OAD (and bend angle)

increases. Base parabola radius of curvature is 20mm. Entrance pupil diameter is 10mm.

Field angle is 0.1 degrees...... 12

Figure 3: OAP rayfan and aberration with respect to parent optic ...... 13

Figure 4: OAP Specification table [2]...... 14

Figure 5: Process for developing an alignment plan and building an optical system [3] (excerpt

from Guyer 2001)[4]...... 16

Figure 6: Global and local coordinate frames for positioning optic ...... 18

Figure 7: Laser tracker viewing an SMR. Image courtesy of FARO Technologies...... 19

Figure 8: SMR in nest. Right displaying corner cube, left tooling ball side [3]...... 20

Figure 9: Bridge model CMM (Mitutoyo)...... 22

Figure 10: Arm model CMM (FARO)...... 22

Figure 11: Fizeau Interferometer setup for an on-axis optic. Image courtesy of Dr. James Wyant

...... 24

Figure 12: Physical setup of an interferometer (Zygo)...... 24

Figure 13: Top view of optical table for vertical alignment [7]...... 27

Figure 14: Side view of optical table for vertical alignment [7]...... 27

Figure 15: Laser tracker comparison chart [8]...... 30

Figure 16: Laser tracker measurement of a 1.7m diameter off-axis aspheric mirror agrees to 0.5

um rms with data from an interferometer. The low order terms of power, astigmatism and

coma, which are strongly affected by alignment, were removed from this data. [8] ...... 31

5 Figure 17: The laser tracker will accurately measure concave surfaces if the tracker is located

near the center of curvature. [8] ...... 32

Figure 18: Increase in published papers for alignment of OAPs ...... 37

6 ABSTRACT

The off-axis parabola (OAP) optical mirror, while simple and elegant in its conceptual design and use, has its complexity embedded within the manufacturing and placement within an optical system. There are several alignment techniques available that can be used to position an

OAP to a level of fidelity required for a given system performance. The applications of OAPs are endless: telescopes, spectrometers, cameras and other instruments. Under these systems, the level of alignment is driven by the precision required by the system needs. This report will present a subset of the techniques and instruments that are currently available to perform the alignment of an OAP and provide the level of precision capable with each method.

7 1 INTRODUCTION

1.1 MOTIVATION

Optical alignment is a critical step in meeting the desired performance specifications of an optical system. Optical systems can range from simple two-lens children’s telescopes to complex high-performing micro-lithography systems. Many advanced systems trying to achieve minimum

(or no) obscuration and/or compact system size include an optical component known as the off- axis parabola or OAP. Often times, the OAP is considered ‘tricky’ to align given its off-axis nature, which means no optical vertex point is on the OAP surface. Given the increasing demand for precision placement of optical components and OAP’s in modern optical systems, a variety of tools and methodologies have been developed which can achieve these critical alignments. This report will touch briefly on the available tools for precision placement of OAP’s and summarize a subset of available methodologies for aligning the OAP.

1.2 THE OAP

An OAP is an off-axis portion of a parabola that has either been cut from a parent optic, or manufactured to the prescription as if it were cut from a parent optic. Optically, an OAP will generate a diffraction limited image at the parent focal point given an input of a collimated beam.

Conversely, an OAP will generate a collimated beam or “plane wave” if the input light is located at the parent focal point.

Figure 1 illustrates a full parabola with two OAPs (shown as grayed-in portions) that have been generated from a parent optic. The distance of which the OAP is located from the center line of the parent parabola is referred to as the off-axis distance (OAD). The OAD defines the centerline to which the OAP has been generated, this is referred to as the optical centerline (OCL) for the

8 off-axis segment of the parent parabola. It is important to note that, provided a spherical wavefront at the focal point of the OAP, the collimated output angle will be independent of orientation of the cone-beam spherical input wavefront. Finally, the off-axis angle or “deviation angle” is the angle defined by the OCL and parent prescription as shown in Figure 1. This angle allows an optical designer the flexibility to place an OAP to accommodate space constraints within a system while maintaining optical performance.

OAPs are typically coated with a reflective coating and used where space is at a premium, the system is not axially symmetric, or the optical design requires the collimating properties of the

OAP. A few common examples of such systems include spectrometers, astronomical telescopes, optical testing collimators, and optical relay systems.

Figure 1: OAP diagram from Optikos [1].

9 Due to the off-axis nature of the OAP, careful considerations must be made when designing, optimizing, and tolerancing the optic for practical application. The OAP must be evaluated for physical placement in all six degrees of freedom (rotation, tip/tilt, piston, and x/y translation), defining its unique position-and-orientation in the entire optical system to ensure final system performance.

1.3 ABERRATIONS OF THE OAP

Aberrations from a parent parabola versus the off-axis parabola are related. The table in

Figure 2 illustrates how the dominant aberration present in the image plane for a non-zero field angle changes from coma to astigmatism as the OAD increases, i.e. from “parent” to OAP. This is a consequence of the geometry of the optic and how an aberration is defined about the image plane

(see Figure 3). For the purposes of this general illustration, model specifications have been chosen arbitrarily to show the overall trend of the aberrations of the OAP.

Off- Ray Diagram Pupil Map OPD plot axis Angle (degree) 0 WAVEFRONT ABERRATION New lens from CVMACRO Waves 1.0000

0.5000

13:28:00 0.0000 Field = ( 0.000,0.1000) Degrees Wavelength = 587.6 nm Defocusing = 0.000000 mm WAVEFRONT ABERRATION

New lens from CVMACRO:cvnewlens.seq Waves 9 1.0000

0.5000

0.0000 Field = ( 0.000,0.1000) Degrees Wavelength = 587.6 nm Defocusing = 0.000000 mm

10 13:51:40

WAVEFRONT ABERRATION

New lens from CVMACRO:cvnewlens.seq Waves 18 1.0000

0.5000

13:53:37

0.0000 Field = ( 0.000,0.1000) Degrees Wavelength = 587.6 nm Defocusing = 0.000000 mm WAVEFRONT ABERRATION New lens from CVMACRO:cvnewlens.seq 27 Waves 1.0000

0.5000

13:56:09

0.0000 Field = ( 0.000,0.1000) Degrees Wavelength = 587.6 nm Defocusing = 0.000000 mm WAVEFRONT ABERRATION New lens from CVMACRO:cvnewlens.seq 36 Waves 1.0000

0.5000

13:58:03 0.0000 Field = ( 0.000,0.1000) Degrees Wavelength = 587.6 nm Defocusing = 0.000000 mm WAVEFRONT ABERRATION

New lens from CVMACRO:cvnewlens.seq Waves 45 1.0000

0.5000

14:01:40 0.0000 Field = ( 0.000,0.1000) Degrees Wavelength = 587.6 nm Defocusing = 0.000000 mm WAVEFRONT ABERRATION New lens from CVMACRO:cvnewlens.seq 54 Waves 1.0000

0.5000

14:05:00 0.0000 Field = ( 0.000,0.1000) Degrees Wavelength = 587.6 nm Defocusing = 0.000000 mm WAVEFRONT ABERRATION New lens from CVMACRO:cvnewlens.seq 63 Waves 1.0000

0.5000

14:14:25 0.0000 Field = ( 0.000,0.1000) Degrees Wavelength = 587.6 nm Defocusing = 0.000000 mm WAVEFRONT ABERRATION New lens from CVMACRO:cvnewlens.seq 72 Waves 1.0000

0.5000

0.0000 Field = ( 0.000,0.1000) Degrees Wavelength = 587.6 nm Defocusing = 0.000000 mm

11 14:18:27

WAVEFRONT ABERRATION New lens from CVMACRO:cvnewlens.seq 81 Waves 1.0000

0.5000

0.0000 14:23:13 Field = ( 0.000,0.1000) Degrees Wavelength = 587.6 nm Defocusing = 0.000000 mm WAVEFRONT ABERRATION New lens from CVMACRO:cvnewlens.seq 90 Waves 1.0000

0.5000

0.0000 Field = ( 0.000,0.1000) Degrees Wavelength = 587.6 nm Defocusing = 0.000000 mm

Figure 2: System diagram, pupil map, and ray-fan for an arbitrary OAP as OAD (and off-axis angle in Figure 1) increases. Base parabola radius of curvature is 20mm. Entrance pupil diameter is 10mm. Field angle is 0.1 degrees.

The rays contributing to the ray fan of the OAP can be thought of as a subset of the rays that would have made up the ray-fan of the full on-axis base parabolic reflector shown in the top row in Figure 3. Considering only those off-axis rays is equivalent to considering a portion of the ray-fan. The table in Figure 3 illustrates this concept in stages using an arbitrarily chosen field angle of 0.1 degrees. At first, the pupil covers a large symmetrical portion of the base parabola that includes the off-axis portion that will be used. Coma is the dominant aberration as shown in the ray-fan and pupil map diagrams (top row in Figure 3). In the 2nd row, the off-axis portion is highlighted in the mirror diagram, the ray-fan, and the pupil. The next step is to tilt the image plane so that it is normal to the ray at the center of the highlighted ray bundle. This will be the new chief ray for the OAP system. Finally, the base parabola is replaced with the off-axis portion. The pupil analysis of this OAP matches the highlighted portion of the overall base parabola pupil. The entire step-by-step process to illustrate the OAP associated aberration is summarized and presented in

Figure 3.

Figure 3: OAP configuration, rayfan and aberration with respect to the parent optic.

13 1.4 SPECIFYING THE OAP

Many vendors are capable of generating and providing a high-performance custom OAP

(SORL, L3, Coherent, and more). Depending on the application, an “off the shelf” OAP from sources like Newport, Thorlabs and Edmund Optics will suffice if designed into the system and meets the system-level performance requirements. In cases where a specific prescription is required the following guidelines in Figure 4 will be required for describing an OAP to a custom

OAP manufacturer.

Specification Description Zero thermal-expansion ceramic (Zerodur, ULE or its equivalent) is the Substrate Material most stable and standard. Alternatives include Fused Silica, conventional metals, and light weighted substrates. Focal Length (FL) True parabolic or vertex focal length Specified along a perpendicular from optical axis to the center of OAP Off-Axis Distance (OAD) mirror surface Outer mirror diameter measurement compatible with physical Outer Diameter (OD) requirements Clear Aperture (CA) The optically qualified region of the OAP mirror surface. To insure structural stability, a mirror will typically be made from a blank having a 6:1 diameter-to-thickness ratio. For stand-alone mirrors, this is Edge Thickness (ET) approximately correct. Standard OAPs sectioned from a larger diameter parent will be thicker Deviation from perfect paraboloidal shape, specified as “peak-to-peak” (peak-to-valley) value in fractions of the interferometric test wavelength Surface Accuracy 0.6328 microns. Due to reflection at the surface (i.e., double-path), the wavefront error is twice the surface error. A measure of the optical polish, specified by “scratch/dig” values denoting surface imperfections, as defined by Military Specifications, MIL-0-1383. Typical Scratch/Dig specifications: Surface Quality 80/50 – Standard IR Quality 60/40 – Visible to IR 40/20 – UV to Visible Laser 20/10 – UV and High-Power Laser May be selected from a variety of evaporated metals and dielectric materials for optimal performance in the wavelength region, optical Optical Coatings power level, and environment of intended operation. Examples of coatings are: Protected Gold, Enhanced Silver

Figure 4: Exemplary OAP Specification table [2].

Additional considerations should be made when requesting and procuring a customized

OAP. A vendor will have the capability of adding additional features (e.g., fiducials) to the optic to aide in the positioning of the optic within a system. Additional features may include mechanical datums that may be accessed using a touch probe such as a coordinate measurement machine

(CMM). A convenient set of physical datums, for example, would generate a coordinate frame to which all relevant optical data could be referenced and aligned with respect to it. With knowledge of the tolerance stack up, positioning the optic to within 100 microns becomes easily achieved.

2 GENERAL ALIGNMENT

Optical alignment is a highly iterative process that requires a well-developed systematic alignment plan to satisfactorily align an optical system given the application. A plan would consider the metrology tools, techniques, data reduction and analysis algorithms [3] and would be evaluated for flow, feasibility (cost, schedule, technique), and consistency. Using a robust approach to developing an alignment plan allows for critical elements, such as manufactured optics and alignment tools, to be considered in order to ensure form and function compliance to levied requirements. Other considerations may include: accessibility, additional tooling, optical support equipment (such as a Computer-Generated Hologram (CGH) and additional optical components such as penta-prisms or lenses), as well as dynamic considerations (such as environmental vibration or temperature sensitivities).

Figure 5 illustrates the iterative process that should be taken when developing an alignment plan. This flow ensures that critical considerations are captured and realistic accommodations can be implemented prior to aligning an optical system.

Figure 5: Process for developing a systematic alignment plan and building an optical system [3].

A well-equipped laboratory for OAP alignment would include measurement tools such as a

CMM, lasers, a laser tracker, an interferometer, a point source microscope, tooling balls, high quality flat mirrors, a tape measure, a protractor and a variety of miscellaneous mechanical mounts and stages. Some of these tools are more sophisticated in their operation than others. Uses for some of the simpler hardware include; the tooling ball: used to constrain the location of the OAP focus, the flat mirror: used to return the light off collimated light generated by an OAP in double- pass. The operation and use of these of some of the more complex tools will be described below, with specific use cases of a subset of these tools discussed in Sections 3 and 4. Alignment tools can be roughly categorized into three main areas: XYZ coordinate measurement instruments,

16 angular measurement instruments [3], and finally optical measurement instruments. This section is not intended to be a comprehensive list of alignment tools and their uses, but rather a general subset one would consider useful in the alignment of the OAP and will focus on the use of coordinate and optical measurement instruments.

2.1 XYZ COORDINATE MEASURING INSTRUMENTS

This section will describe two major types of instruments typically used in the placement (with acquisition of positional knowledge) of hardware and optical components. To begin aligning an optical system, knowing where each of the components needs to be placed with respect to a coordinate frame is very helpful and likely part of the development flow illustrated in Figure 5. A coordinate frame can be defined using a physical datum structure or optical axis construct to which optical elements will be placed. Additionally, the coordinate system can also be used at the component level for generating the optic itself and relating critical optical prescriptions to a set of physical features/fiducials located on the optic. Such features can be in the form of accessible physical forms such as holes, alignment balls, and flat surfaces which can then be measured using a laser tracker or CMM. These features can be singular points to define an origin, a set of flats to define XYZ planes or a combination of points and flats.

Characterization of alignment features used to establish the coordinate frame must be stable and well calibrated under the environmental conditions which they are to be used to reduce errors that may contribute to overall alignment error. Coordinate systems are critical to ensure that all subsystem components will line up as expected. In the case of the OAP, a well-established local coordinate frame will provide knowledge of the focus location and orientation of the optical prescription in the context of the physical OAP specified in Figure 6.

Figure 6: Global and local coordinate frames for positioning OAP mirror in an optical system

2.1.1 The Laser Tracker

A laser tracker (LT), shown in Figure 7, can be used to scan a surface to measure a plane or curved shape using the signal return from a spherically-mounted retroreflector (SMR) in contact with and dragged (or discretely sampled) along the surface of interest [3]. SMR’s can also be used in the initial assessment of an optical component to characterize critical optical parameters with respect to the local coordinate frame of the optic as depicted in Figure 6. The data format of a laser tracker (as well as the CMM described later) will generate a set of point clouds that are later evaluated against an ideal model for positioning of components with respect to a global coordinate

18 frame. Angle measurements can be also made or derived from the point data results of some of these instruments, however, the uncertainties of angular data typically are much higher than instruments that are specifically designed for angular type measurements [3] such as autocollimators and alignment telescopes.

Figure 7: Laser tracker system viewing an SMR. Image courtesy of FARO Technologies.

The laser tracker is an optical device which uses angular azimuth and elevation encoders to determine the range of angular movement of a target. There are two operating modes of a laser tracker acquiring the distance to the target (given a particular model and its capability) which are

1) interferometric relative distance (DMI) and 2) absolute distance metering (ADM). The ADM method is considered the most flexible and is easiest to use because the tracker can look from one target to the next [8] without worrying about breaking the laser beam path between measurements.

On the other hand, the DMI is more accurate in that it uses interferometry, but is limited in its ease of use to due susceptibility to measurement errors from index variation of the media (e.g., air).

19 Also, the requirement of not-breaking the beam paths throughout the entire measurement sequence often becomes a practical challenge with line of site and accessibility often being challenging for uniquely designed systems. Measured motions of a target are translated into a defined coordinate frame by the operating software whereby a point cloud can be generated of a system. A target is a SMR (Spherical Mounted Retroreflector), also known as a corner cube mounted within a high precision tooling ball shown in Figure 8. The precision of the SMR also lies in the coincidence of the corner cube apex with the center of curvature of the tooling ball. Typical offsets of the apex to curvature are on the order of roughly 4 microns for a 1.5” diameter SMR [5]. The advantage of an

SMR is that it returns an incoming ray of light with no (or very little) angular deviation to the original source depending on the quality of the corner cube. The SMR is held against the object under measurement (e.g., OAP mirror) at multiple sampling locations using highly repeatable nesting mounts. These mapped locations build a virtual model of the object that can be used as a reference for subsequent measurements. This is useful when needing to position an OAP within the capability of the laser tracker.

Figure 8: SMR in nest. Left displaying corner cube, right tooling ball side [3].

The capabilities, accuracy, and repeatability of a laser tracker is dependent on the manufacturer, and model of the equipment. Measurement performance will degrade as a function of size or distance of the object and distance of the laser tracker to the object. Errors in

20 measurements can be reduced by taking many measurements and averaging the measurement values and errors. Typical LT measurement uncertainties are approximately 10-25 microns (1- sigma; accuracy) per meter of range along the horizontal and vertical directions, driven by uncertainty in the angular encoders, and about the same uncertainty in the range direction, but about constant in magnitude until factors like air turbulence and temperature uncertainty become important with increasing range [3].

A recent search into commercially available laser trackers from FARO Technologies indicated the following capability for their “ION” model; Distance measurements: resolution of 50 µm volumetric accuracy at 10 meters with accuracy of 8 µm + 0.4 µm per meter, Angular measurements: angular accuracy 10 µm + 2.5 µm per meter, with a precision accuracy of 2 arc seconds. Additional laser tracker models and technologies are described and presented in

Section 3.2.

2.1.2 The Coordinate Measurement Machine (CMM)

The CMM comes in a variety of sizes and accuracies and is dependent on the availability, budget, and mechanical configuration of the object under test. As with the laser tracker, a CMM uses encoders in conjunction with precision stages to manipulate a probe within a volume limited by the CMM model. There are two types of CMM’s, contact and non-contact, but for purposes of this report contact CMMs will be discussed. The measurement accuracy of a CMM can be as low as 0.28 microns (Mitutoyo) over a limited working volume which degrades per linear distance from the defined working volume. These contact probes also come in a variety of configurations such a ‘bridge’ model or ‘arm’ model as shown in Figures 9 and 10, respectively.

Figure 9: Bridge model CMM (Mitutoyo).

Figure 10: Arm model CMM (FARO).

22 A CMM can be used to position an optic in all six degrees of freedom. The position accuracy being limited by the capability of the instrument and the geometry of the optic itself. The operation of the CMM is that a physical probe is used to physically touch an optic (though there are models that use a non-contact optical probe, but that will not be discussed here). Using the same approach as the Laser Tracker, encoders track the position of the CMM and probe to build a virtual 3- dimensional point cloud model of the object.

Similar to that of the laser tracker, a commercially available CMM from FARO technologies for the “GAGE” model indicated that within 100 mm working distance of the arm, the accuracy is

5.8 µm, with performance decreasing to 14.6 µm at 1200 um from the arm (Figure 10).

2.2 THE INTERFEROMETER

The interferometer comes in a variety of configurations to collect data of the optical performance in terms of wavefront quality for an optical component or system. The fundamental operation of an interferometer involves a coherent light source generating a wavefront that is separated into two parts: a) reference wavefront, b) measurement wavefront. The measurement wavefront is propagated to the optical system under test and returned to the interferometer. The measurement and reference wavefronts are then recombined through a process called

“superposition” to generate interference fringes according to the optical path difference between them. These fringes enable an operator to view the wavefront aberrations due to any system misalignment within the optical system and thereby enable corrective alignments to be taken to reduce the magnitude and type of aberrations.

A typical interferometer found in most labs is the Fizeau Interferometer with a reference sphere generating a diverging spherical test wavefront as shown in Figure 11. For example, an

23 interferometer (e.g., Zygo dynafiz in Figure 12) will perform a phase-shifting in order to determine, through the phase-shifted multiple fringe patterns, whether a surface sag deviation on an optic is a “high” or “low” and measure the wavefront information for the optical system or surface under test.

Figure 11: Fizeau Interferometer setup with a reference sphere for an on-axis concave spherical mirror metrology. Image courtesy of Dr. James Wyant course notes

Figure 12: Physical setup and configuration of an interferometer measuring optical flat on an optical table (Zygo dynafiz).

24 3 OAP SPECIFIC ALIGNMENT TECHNIQUES

3.1 TWO LASER TECHNIQUE

The motivation for developing the two-laser technique originated from the limitation of obtaining high-quality refractive lenses at sizes that could be used for large OAPs up to 80 inches in diameter and larger. The two-laser technique employs two HeNe lasers that are positioned parallel to each other. This parallel pair are then positioned along the optical axis of the OAP. The focal point of the OAP is automatically found where the two reflected beams cross each other [7].

The alignment procedure is rather methodical in approach, in that the beams are optimized independently to position the optic in vertical and horizontal orientations.

The alignment procedure begins with the initial setup of two parallel laser beams. A HeNe laser is split into two separate beams via a beamsplitter. The separation between the beams is determined by the clear aperture of the optic. An iris is moved along each beam to compare and confirm the beam height along the optical table. Mirrors are employed to adjust the beam height through tilt. Multiple techniques can be used to adjust the parallelism of the beams, and in this case the distance between the beams, ∆�, is evaluated along the optical table using a ruler and the tip of the mirrors are adjusted to adjust the parallelism. Figure 13 illustrates the overall setup configuration of the two-laser technique aligning an OAP.

The following steps are a direct quotation of reference [7] and will align the parabola vertically:

1) The off-axis parabolic mirror is first put on the optical table such that its optical axis is

nearly parallel to the incident beams. We call the incident beams, beam 1 and beam 2,

where beam 1 is farther away from the optical axis than beam 2 (see Figure 13).

25 2) Then we make reflected beam 1 parallel to the surface of the optical table by rotating the

mirror about the x axis. To check parallelism, we monitor the height of beam 1 with the

same iris that is used to make the incident beams parallel. Since incident laser beams have

nonzero beam-widths, by the OAP mirror surface curvature, reflected beam 1 first

converges to a small spot and then diverges. We put the iris at the position where reflected

beam 1 becomes slightly larger than the opening of the iris and compare the beam spot

with the iris opening.

3) Next, we monitor height of reflected beam 2 with the same iris to see if the beam is parallel

to the surface of the optical table. In this case the same method is used as for reflected

beam 1 described in Step 2.

4) If reflected, beam 2 propagates upward, we decrease the height of the off-axis parabolic

mirror.

5) Similarly, if reflected beam 2 propagates downward, we increase the height of the mirror

as shown in Figure 14 until the OAP is vertically aligned with respect to the beam 1 and 2.

As mentioned before, this process is highly iterative, and steps 2 through 5 are repeated until the beams 1 and 2 are made parallel to the optical table. This will put the optical axis of the off- axis parabola in the plane made by the two incident beams and make the y-axis perpendicular to the surface of the optical table [7] (i.e. the optic is now aligned vertically).

Figure 13: Top view of the optical table configuration using two parallel laser beams for vertical

alignment of OAP [7].

Figure 14: Side view of the optical table configuration using two parallel laser beams for

vertical alignment of OAP [7].

27 To align the horizontal axis of the OAP, a flat mirror is positioned at the intersection of the two beams from the prior vertical alignment. The following steps are then iterated for completing the alignment process [7]:

1) We tilt the plane mirror to send one of the beams (either beam 1 or 2) to a point at the off-

axis parabolic mirror, which is between and above the two incident beams. This beam is

called beam 3. We find the crossing point of the reflected beams by observing scattering

of the beams from the surface of the plane mirror. By chopping one of the incident beams

with a piece of paper we can easily see how well the beams overlap at the surface of the

plane mirror.

2) Next, we check the height of the reflected beam 3 to see if the beam is parallel to the surface

of the optical table. This is done with another iris diaphragm, which can be moved on the

table with a fixed height. The height is checked at two different locations on the table by

changing the size of the iris opening because beam 3 diverges slightly after reflecting from

the off-axis parabolic mirror.

3) If the reflected beam 3 propagates upward, we rotate the off-axis parabolic mirror about

the y-axis in the direction of smaller angle of incidence for beams 1 and 2.

4) Similarly, if the beam propagates downward, we rotate the mirror in the direction of larger

angle of incidence.

5) Steps 1 through 4 are repeated until reflected beam 3 is parallel to the surface of the optical

table. During iteration of the procedure the plane mirror is fine adjusted to send beam 3 to

the same point at the off-axis parabolic mirror. This is to observe an angular change of

reflected beam 3 in a consistent manner for a small rotation of the off-axis parabolic mirror.

When the parabolic mirror is aligned such that reflected beam 3 is parallel to the surface of

28 the optical table, the optical axis becomes parallel to the incident beams 1 and 2, and the

focal point is found at a point where reflected beams 1 and 2 cross each other.

Through comparison with another technique (i.e., using a point source microscope) and analytically evaluating the alignment errors, this showed that locating the focal point for the two- laser technique was off by roughly 3 mrad vertically and horizontally. This alignment technique relies on vendor knowledge of the optic and the coordinate frame originally established.

3.2 LASER TRACKER TECHNIQUE

The “Laser Tracker Technique” evokes the capabilities described in Section 2.1.1 The

Laser Tracker of laser tracker (LT) capabilities. SMR references are placed on the OAP during the manufacturing process, which relates all pertinent optical details of the OAP to a local coordinate frame established by a set of fiducials. However, a single SMR is not adequate to define the position of an optic. Specifically, an OAP requires all 6 degrees of freedom be constrained and surveyed with respect to a reference coordinate system, which would require more than 1 SMR be used in establishing the local optical coordinate frame (ideally three or more

SMRs). A reference coordinate system is useful when placing an optic in an optical system with other components, mechanical or optical as illustrated in Figure 6.

The placement of an OAP takes advantage of two operating modes of the LT: Absolute

Distance Measurement (ADM), whereby the absolute distance to the SMR touching the optic is measured, or Distance Measuring Interferometry (DMI), where the relative change in a distance measurement is observed as the SMR moves from a known initial location to the optic under measurement. The DMI method is much more accurate that the ADM approach because DMI

29 employs interferometric approach as evidenced in the summary chart comparing a few commercial laser trackers in Figure 15. The performance is exceptionally good for a system that requires micron level accuracy and sub-arcsec resolution. However, as mentioned by the authors [8], these systems are highly susceptible to small problems such as the refractive index of air, which can cause variability in the measurements.

Figure 15: Commercially available laser tracker system comparison chart [8].

For measurements of the OAP mirror surface, the LT is place at the center of curvature of the optic and an SMR is scanned across the surface of the optic. A comparison of LT measurements versus an interferometric measurement yielded approximately 25% difference in surface measurement after subtracting the alignment-related terms as presented in Figure 16.

Figure 16: Laser tracker measurement of a 1.7m diameter off-axis aspheric mirror agrees to 0.5 µm rms with data from an interferometer. The low order terms of power, astigmatism and coma, which are strongly affected by alignment, were removed from this data. [8]

This indicates that the method for placement of optic using a laser tracker can yield results that may be “good enough” for systems that may not require the fidelity of an interferometer. If there are a minimum of three SMRs located accurately on an optic referencing the prescription, then positioning (including the orientation) the optic to LT accuracies can be achieved [8], with surface figure data included as well unlike the Two Laser technique in Section 3.1.

The following is an excerpt from reference [8]: By carefully controlling the geometry, the LT can measure aspherical optical surfaces to < 1 µm accuracy, with additional accuracy supported by additional calibration. To accomplish these measurements, the LT is supported near the center of curvature of the concave surface and the SMR is scanned across the surface. It is important to note, that having the tracker near the center of curvature, the angular uncertainties couple weakly into the measurements, thereby improving the overall error. The radial direction is measured to high precision using the DMI mode, which provides accuracy of << 1 µm in the radial direction, which nearly matches the surface normal. For the case where the tracker is located a distance h from the center of curvature of the mirror under test, the sensitivity of the radial measurement to angular measurement errors is derived as:

31 ,

Equation 1: Sensitivity of radial measurement to angular measurement errors [8].

where x is the off-axis distance for the point being measured and R gives the nominal radius of curvature.

Figure 17: The LT will accurately measure concave surfaces if the LT is located near the center of curvature as the angular encoder value uncertainties couple weakly into the measurements. [8]

To align an OAP, the use of this LT scanning method will provide knowledge of the surface figure error (SFE) of the OAP as well as be able to relay that information to a set of fiducials that can be referenced while aligning the OAP within the optical system.

32 3.3 COMPUTER AIDED TECHNIQUE USING ZERNIKES

Zernike coefficients describe the aberration makeup of an optical system, or in our case a mis- aligned optic. There are several Zernike definitions which include the ANSI standard scheme, the

Noll scheme and of course, Wyant’s scheme. For convention, this paper will reference the Noll

Zernike scheme, see Appendix A for the Noll Zernike numbering map. The method described here minimizes the merit function of Zernike coefficients rather than using the sensitivities of the coefficients [9]. Using sensitivities showed a decrease in alignment sensitivity as alignment errors increased. This is due to the Zernike coefficient to misalignment parameters not being sufficiently linear or one or more misalignment parameters coupling with each other [9]. Therefore, several iterations are required to achieve optimal alignment.

The Merit Function (MF) approach considers three main factors to compute the alignment: the number of Zernike coefficients, the field error, weighting factors of each parameter [9] and as-built data of the optic. Accuracy of the MF function can be improved by employing additional Zernike coefficients, but low order coefficients (e.g., lower than the 9th term) are likely sufficient to align a simple optic or system. Additional measurements and different fields may be necessary to reduce errors from ambiguity in Zernike coefficient makeup and to reduce impacts from environmental effects such as turbulence and temperature. However, for purposes of aligning an OAP, a singular field point with the Zernike coefficients will suffice.

The MF approach for alignment uses the following definition for the MF:

Equation 2: Merit function definition for most optical design software [9].

33 The following is an excerpt from reference [9]: � is the current values of each compensator,

� are the target values, and � are the weighting factors. The optical design software such as

ZEMAX or CODE V usually minimizes the MF value to find a potential solution. To determine the misalignment of the system, the V’s and T’s are the values of Zernike coefficients which are minimized through the embedded algorithm of the optical design software package. For example,

ZEMAX uses the actively damped least square method to minimize the MF. In order to efficiently apply this method, a coded macro program is authored to carry out the following functions:

1. Read the measured Zernike coefficients from the interferometric measurement (this is the

target value).

2. Record and allocate of each Zernike coefficients as currently measured into the target

column in MF editor.

3. Optimize the system using an embedded algorithm on the optical design software. The

number of optimizations can be adjusted.

4. The result is the misalignment of each compensator (defining the position and orientation

of the optic under alignment), which should be compensated.

Theoretical performance indicates that through two iterations of the measurement to compensation method yielded a fully optimized, perfectly performing system with no errors.

In conventional alignments, the use of the Zernike coefficients required that the original system alignment was “close enough” and not dominated by alignment errors. While this technique was originally explored to align a specific telescope, this approach can be extended to single elements such as the OAP.

The predicted theoretical performance is zero across all degrees of freedom. However, in reality, due to the noise and uncertainty sources, there are always practical limits and errors

34 depending on the noise level. The MF method offers additional control of an optic or optical system across multiple fields relatively quickly and more accurately than by pure sensitivities alone.

Placement of an OAP can take advantage of the knowledge of the low order Zernike coefficients as long as the position of the OAP can be referenced within a coordinate frame (otherwise simply placing the optic on a table would be “alignment” enough).

3.4 SUMMARY

Three unique alignment techniques were studied and discussed to illustrate the variety of alignment methods available to align an OAP. Specific considerations to system performance are required in order to best select the appropriate technique. Main drivers are cost, schedule, capability and available instrumentation, and personnel expertise. While all techniques can be considered equivalent in schedule, cost and capability will be the main drivers (as shown in

Table 1) for selecting the appropriate approach.

Estimated Method Equipment Capability Cost* Two Laser Technique [2] 2 × HeNe Laser ~$2,000 Resolution: 3 mrad in tip, tilt Laser Tracker Resolution 0.00034 mrad Specialized Laser Tracker Technique [3] ~$100,000 Repeatability: 2.5 µm/m Software Accuracy: 5 µm/m SMRs Interferometer Computer Aided Technique using Specialized ~$60,000 Theoretically perfect alignment Zernike coefficients [4] Software * All techniques assume access to a computer with MATLAB and CodeV/Zemax and an optical table. Cost of table, OAP and OAP mount/adjustment parts are not included.

Figure 18: Executive summary comparison of three different OAP alignment techniques.

35 4 CASE STUDY

When initially approaching the alignment of any optic, the approach this author prefers to use is understanding what degrees of freedom (DOF) need to be constrained in order to achieve optical alignment. Additionally, an optic cannot be placed unless specific requirements have been levied and understood. Not all optical systems require a rigorous approach to alignment, and based on performance needs and often times, budgetary constraints, the level to which all DOF’s can be aligned are limited.

Consider a 100-millimeter diameter OAP mirror segment with a 2-meter focal length that is

35-millimeter distance decentered from the parent parabola. As-built data of the optic (tolerances on conic, radius of curvature, decenter, and the like) have already been implemented within a full system model. Additionally, the vendor has provided knowledge of the optical prescription to a set of SMRs on the rear side of the optic. Using this data, the system requires the placement of the optic to be within 50 microns of its prescribed location and 2 microradians in tip and tilt. This particular system to which the OAP is to be installed, is volumetrically challenged and prevents accessibility to the front surface of the optic. The choices for such an alignment decreed using a laser tracker. Our choice of laser trackers to fulfill the job is abundant, as indicated in Figure 15, with the FARO model being an excellent high precision choice.

A literature search into the alignment of OAPs within SPIE’s database indicated over 550 papers at the time of authoring this report (December 2018). Of the 550 papers, over 400 were associated with alignment of OAPs within telescopes, with 84 papers coming from the Jet

Propulsion Laboratory, and almost 50 papers written for the James Webb Space Telescope

(JWST). Interestingly, there’s been a linear increase over the last 30 years of published papers involving OAP alignment as indicated by Figure 19.

140

120

100

SPIE Published Papers (#) SPIE Papers Published 40

0 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 Years

Figure 19: Increase in published papers for alignment of OAPs based on the SPIE digital library database

A brief look into the alignment techniques employed by this research, indicated the need for precision alignment of these telescopes using wavefront sensing and control. Over 260 papers were written involving wavefront sensing, indicating that the prevalent approach is through merit functions or system sensitivities. However, it is not clear if wavefront sensing used system sensitivities or direct merit function (discussed in Section 3.3) were used to align the systems.

Other prevalent alignment techniques utilized the initial placement of an OAP using a laser tracker and employing an interferometer to finely align the system.

37 5 CONCLUSION

The motivation behind this report was based a on a real world need for aligning a very specific system which consisted of an OAP mirror. Most of the techniques explored in this paper will be utilized in a consolidated fashion for the high precision and knowledge that is required (i.e., CGH and laser tracker approaches) as evidenced by the brief look across SPIE publications with OAP use cases. However, most laboratories are equipped with an interferometer and a standard computation software package (such as MATLAB and Zemax), thereby making the Zernike coefficient-based alignment approach a convenient way to precisely position and orient an OAP mirror. Should the novice or budget constrained metrology approach be required, the relatively inexpensive approach of the two-laser technique may be viable if system parameters can be met.

38 APPENDIX A – Noll Zernike Expansion

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